Mr. Ivory's Tlieory of the Velocity of Sound. 255 



sufficient for finding, generally, the relation between the den- 

 sity and latent heat, when these quantities vary together. 



It must not, however, be imagined that the damage arising 

 from the inadvertency that has been noticed, is ruinously great. 

 The formulae obtained are true to quantities of the second 

 order with respect to « and /3. They are sufficiently exact 

 for investigating the velocity of sound ; and they can hardly 

 lead to any error of moment in any practical inquiry. But it 

 is always best to square our speculations according to expe- 

 rience and the laws actually followed in nature ; and, in a case 

 like the present, when it may be supposed that we have re- 

 turned into the right path after having deviated a little from 

 it, it is instructive to look back and examine what led us 

 astray. 



In further illustration of what has been said, it may not be 

 improper to add a few words concerning the equations in the 

 xiith book of the Mecanique Celeste, pp. 123, 127. For this 

 purpose I seek the values of x and i from the foregoing equa- 

 tions (C); then, by taking the sum, we get, 



.H.,=V = (i.i-l)l±^+(f-l)-'i^- 

 Put A- = 1 + — as before, and differentiate : then 



d V . , d V „ \ -\- aS { P — P' 



!7 



dV , d\ 



We have initially, p = 'p', ^ = ^; and if we suppose that the 

 mass of air undergoes only a small variation from the initial 

 state, we shall have, 



e + /c -J— p = 



dV 



I _ ^g I 



Up P 

 These equations are true only at one point, and in a particu- 

 lar state of the variables, as has been mentioned. They can 

 have nothing to do with integration, which supposes that the 

 differential equations are exact for all values of the flowing 

 quantities within the limits of their variation. They merely 

 express that the two specific heats, under a constant pressure 

 and under a constant volume, have to one another the same 

 invariable proportion, whatever be the condition of the mass 

 of air. 



March 5. 182?. J- I^^^^^" 



LI 1 1. Theory 



